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G.C.
G.C.
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They all have cohomological dimension $2$.

The upper bound follows easily from the Serre spectral sequence, the lower bound follows from the fact that groups of cohomological dimension $1$ are free groups. This last fact can be proved without using this big theorem by computing $H^*(G,\mathbf ZG)$ : it is concentrated in dimension 2 and is a free abelian group ... in particular your groups are duality groups but(but rarely Poincaré duality groups).

Edit : sorry, I answered too quickly ... the group $H^2(G,\mathbf ZG)$ does not need to be a free abelian group. Thus your groups need not be duality groups.

They all have cohomological dimension $2$.

The upper bound follows easily from the Serre spectral sequence, the lower bound follows from the fact that groups of cohomological dimension $1$ are free groups. This last fact can be proved without using this big theorem by computing $H^*(G,\mathbf ZG)$ : it is concentrated in dimension 2 and is a free abelian group ... in particular your groups are duality groups but rarely Poincaré duality groups).

They all have cohomological dimension $2$.

The upper bound follows easily from the Serre spectral sequence, the lower bound follows from the fact that groups of cohomological dimension $1$ are free groups. This last fact can be proved without using this big theorem by computing $H^*(G,\mathbf ZG)$ : it is concentrated in dimension 2 and is a free abelian group ... in particular your groups are duality groups (but rarely Poincaré duality groups).

Edit : sorry, I answered too quickly ... the group $H^2(G,\mathbf ZG)$ does not need to be a free abelian group. Thus your groups need not be duality groups.

Source Link
G.C.
G.C.
  • 291
  • 1
  • 4

They all have cohomological dimension $2$.

The upper bound follows easily from the Serre spectral sequence, the lower bound follows from the fact that groups of cohomological dimension $1$ are free groups. This last fact can be proved without using this big theorem by computing $H^*(G,\mathbf ZG)$ : it is concentrated in dimension 2 and is a free abelian group ... in particular your groups are duality groups but rarely Poincaré duality groups).